On the stable manifolds of difference equations with infinite delay

Abstract

This paper is concerned with difference equation $ y(n+1) = A_ny_n+f(n, \, y_{n+\delta_n}, \, \lambda) $ with infinite delay in a Banach space $ X $, where $ \delta_n $ ($ n\in {\mathbb Z} $) is a given sequence taking values in $ \{0, 1\} $, and $ \lambda $ is a parameter. First, we construct a Lipschitz invariant manifold $ {\mathcal M}^{\lambda} $ of the equation which consists of all bounded forward solutions. Then we prove that $ {\mathcal M}^{\lambda} $ contains a unique bounded complete solution $ \gamma^{\lambda} = \gamma^{\lambda}(n) $ ($ n\in {\mathbb Z} $) which depends on $ \lambda $ continuously. To understand the dynamical behavior of the equation on the manifold $ {\mathcal M}^{\lambda} $, we establish a discrete inequality with infinite delay. By applying this inequality we finally show that $ \gamma^{\lambda} $ forward attracts (exponentially attracts) all solutions on $ {\mathcal M}^{\lambda} $ provided that the Lipschitz constant of $ f $ is sufficiently small. Smoothness of the manifold $ {\mathcal M}^\lambda $ is also addressed.